This article discusses the dual interpretation of the Jeffreys-Lindley paradox associated with Bayesian posterior probabilities and Bayes factors, both as a differentiation between frequentist and Bayesian statistics and as a pointer to the difficulty of using improper priors while testing. I stress the considerable impact of this paradox on the foundations of both classical and Bayesian statistics. While assessing existing resolutions of the paradox, I focus on a critical viewpoint of the paradox discussed by Spanos in Philosophy of Science.

Since the choice of a particular loss function strongly influences the resulting inference, it seems necessary to rely on “intrinsic” losses when no information is available about the utility function of the decision-maker, rather than to call for classical losses like the squared error loss. Since this setting is quite similar to the derivation of noninformative priors in Bayesian analysis, we first recall the conditions of this derivation and deduce from these conditions some requirements on the intrinsic losses. It then (...) appears that these loss functions should only depend on the sampling distribution and that they should be independent of the parameterization of the distribution. The resulting estimators are therefore transformation equivariant. We study the properties of two natural intrinsic losses, namely entropy and Hellinger losses, and show that they can be expressed in closed form for exponential families. Moreover, the entropy loss also provides analytic expressions of Bayes estimators under conjugate priors; the derivation of Bayes estimators associated with the Hellinger loss is more cumbersome, as shown in Poisson and Gamma cases, while leading to similar estimators. (shrink)

This paper is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.

According to the fitting-attitude analysis of value , to be valuable is to be a fitting object of a pro-attitude. In earlier publications, setting off from this format of analysis, I proposed a modelling of value relations which makes room for incommensurability in value. In this paper, I first recapitulate the value modelling and then move on to suggest adopting a structurally similar analysis of probability. Indeed, many probability theorists from Poisson onwards did adopt an analysis of this kind. This (...) move allows to formally model probability and probability relations in essentially the same way as value and value relations. One of the advantages of the model is that we get a new account of Keynesian incommensurable probabilities, which goes beyond Keynes in distinguishing between different types of incommensurability. It also becomes possible to draw a clear distinction between incommensurability and vagueness in probability comparisons. (shrink)

Absolute linguistic universals are often justified by cross-linguistic analysis: If all observed languages exhibit a property, the property is taken to be a likely universal, perhaps specified in the cognitive or linguistic systems of language learners and users. In many cases, these patterns are then taken to motivate linguistic theory. Here, we show that cross-linguistic analysis will very rarely be able to statistically justify absolute, inviolable patterns in language. We formalize two statistical methods—frequentist and Bayesian—and show that in both it (...) is possible to find strict linguistic universals, but that the numbers of independent languages necessary to do so is generally unachievable. This suggests that methods other than typological statistics are necessary to establish absolute properties of human language, and thus that many of the purported universals in linguistics have not received sufficient empirical justification. (shrink)

This article explores some of the philosophical implications of the Bayesian modeling paradigm. In particular, it focuses on the ramifications of the fact that Bayesian models pre‐specify an inbuilt hypothesis space. To what extent does this pre‐specification correspond to simply ‘‘building the solution in''? I argue that any learner must have a built‐in hypothesis space in precisely the same sense that Bayesian models have one. This has implications for the nature of learning, Fodor's puzzle of concept acquisition, and the role (...) of modeling in cognitive science. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

If the human race comes to an end relatively shortly, then we have been born at a fairly typical time in the history of humanity; if trillions of people eventually exist, then we have been born in the first surprisingly tiny fraction of all people. According to the 'doomsday argument' of Carter, Leslie, Gott and Nielsen, this means that the chance of a disaster which would obliterate humanity is much larger than usually thought. But treating possible observers in the same (...) way as those who actually exist avoids this conclusion: our existence is more likely in a race which is long-lived, and this cancels out the doomsday argument, so that the chance of a disaster is only what one would ordinarily estimate. (shrink)

It is argued that models of H. Jeffreys' axioms of probability (Jeffreys [1939] 1967) are not monotone even with I. J. Good's proposed modification (Good 1950). Hence the additivity axiom seems essential to a theory of probability as it is with Kolmogorov's system (Kolmogorov 1950).

Gott (Nature 363:315–319, 1993) considers the problem of obtaining a probabilistic prediction for the duration of a process, given the observation that the process is currently underway and began a time t ago. He uses a temporal Copernican principle according to which the observation time can be treated as a random variable with uniform probability density. A simple rule follows: with a 95% probability.

Gambles which induce the decision-maker to experience ambiguity about the relative likelihood of events often give rise to ambiguity-seeking and ambiguity-avoidance, which imply violation of additivity and Savage's axioms. The inability of the subjective Bayesian theory to account for these empirical regularities has determined a dichotomy between normative and descriptive views of subjective probability. This paper proposes a framework within which the two perspectives can be reconciled. First, a formal definition of ambiguity is given over a continuum ranging from ignorance (...) to risk, and including ambiguous contexts as subsets. Second, it is shown that the systems of inductive logic account for the effects of ambiguity. Then, Carnap's X-system is applied as a psychological model and compared to Einhorn and Hogarth's non-normative psychological model. Finally, the implications of this research to the modeling of subjective probability judgements are discussed. (shrink)

Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is (...) the Ockham efficiency theorem, which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their non-Ockham competitors. The theorem neglects, however, to consider competitors following random strategies and in many applications random strategies are known to achieve better worst-case loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the second extension concerns retractions in chance, times of retractions in chance, and chances of errors. (shrink)

This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the constraints on rationally coherent confirmational (...) commitments. In the case where credal judgments are numerically determinate confirmational commitments correspond to Carnap’s credibility functions mathematically represented by so—called confirmation functions. Serious investigation of the conditions under which confirmational commitments should be changed ought to be a prime target for critical reflection. The necessarians were mistaken in thinking that confirmational commitments are immune to legitimate modification altogether. But their personalist or subjectivist critics went too far in suggesting that we might dispense with confirmational commitments. There is room for serious reflection on conditions under which changes in confirmational commitments may be brought under critical control. Undertaking such reflection need not become embroiled in the anti inductivism that has characterized the work of Popper, Carnap and Jeffrey and narrowed the focus of students of logical and methodological issues pertaining to inquiry. (shrink)

According to grounded cognition, words whose semantics contain sensory-motor features activate sensory-motor simulations, which, in turn, interact with spatial responses to produce grounded congruency effects. Growing evidence shows these congruency effects do not always occur, suggesting instead that the grounded features in a word's meaning do not become active automatically across contexts. Researchers sometimes use this as evidence that concepts are not grounded, further concluding that grounded information is peripheral to the amodal cores of concepts. We first review broad evidence (...) that words do not have conceptual cores, and that even the most salient features in a word's meaning are not activated automatically. Then, in three experiments, we provide further evidence that grounded congruency effects rely dynamically on context, with the central grounded features in a concept becoming active only when the current context makes them salient. Even when grounded features are central to a word's meaning, their activation depends on task conditions. (shrink)

Common methods for analysing response time (RT) tasks, frequently used across different disciplines of psychology, suffer from a number of limitations such as the failure to directly measure the underlying latent processes of interest and the inability to take into account the uncertainty associated with each individual's point estimate of performance. Here, we discuss a Bayesian hierarchical diffusion model and apply it to RT data. This model allows researchers to decompose performance into meaningful psychological processes and to account optimally for (...) individual differences and commonalities, even with relatively sparse data. We highlight the advantages of the Bayesian hierarchical diffusion model decomposition by applying it to performance on Approach–Avoidance Tasks, widely used in the emotion and psychopathology literature. Model fits for two experimental data-sets demonstrate that the model performs well. The Bayesian hierarchical diffusion model overcomes important limitations of current analysis procedures and provides deeper insight in latent psychological processes of interest. (shrink)

Over the years several non-equivalent probabilistic measures of coherence have been discussed in the philosophical literature. In this paper we examine these measures with respect to their empirical adequacy. Using test cases from the coherence literature as vignettes for psychological experiments we investigate whether the measures can predict the subjective coherence assessments of the participants. It turns out that the participants’ coherence assessments are best described by Roche’s coherence measure based on Douven and Meijs’ average mutual support approach and the (...) conditional probability. (shrink)

Orthodox Probabilists hold that an inquirer ought to harbor a precise degree of confidence in each hypothesis about which she is concerned. Modest Probabilism is one of a family doctrines inspired by the thought that Orthodox Probabilists are thereby demanding that an inquirer effect a precision that is often unwarranted by her evidence. The purpose of this essay is (i) to explain the particular way in which Modest Probabilism answers to this thought, and (ii) to address an alleged counterexample to (...) Modest Probabilism meant to offer proof that Modest Probabilism is no better than its orthodox predecessor at avoiding unwarranted precision. (shrink)

We consider work extraction from two finite reservoirs with constant heat capacity, when the thermodynamic coordinates of the process are not fully specified, i.e., are described by probabilities only. Incomplete information refers to both the specific value of the temperature as well as the label of the reservoir to which it is assigned. Based on the concept of inference, we characterize the reduced performance resulting from this lack of control. Indeed, the estimates for the average efficiency reveal that uncertainty regarding (...) the exact labels reduces the maximal expected efficiency below the Carnot value ), its minimum value reproducing the well known Curzon–Ahlborn value: \ . We also estimate the efficiency before the value of the temperature is revealed. It is found that if the labels are known with certainty, then in the near-equilibrium limit the efficiency scales as \ , while if there is maximal uncertainty in the labels, then the average estimate for efficiency drops to \ . We also suggest how the inferred properties of the incomplete model can be mapped onto a model with complete information but with an additional source of thermodynamic irreversibility. (shrink)

In Chapter 12 of the General Theory, on “The State of Long-Term Expectation,” Keynes writes: “It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain”. In a footnote to this sentence, Keynes points out that by “very uncertain” he does not mean the same as “very improbable” and refers to the chapter on “The Weight of Arguments” in his earlier Treatise on Probability. The purpose of this article, in the first place, is (...) to provide an account of, and to sort out the relations between, Keynes's views on probability, uncertainty, and the weight of arguments. (shrink)

This paper aims to make explicit the methodological conditions that should be satisfied for the Bayesian model to be used as a normative model of human probability judgment. After noticing the lack of a clear definition of Bayesianism in the psychological literature and the lack of justification for using it, a classic definition of subjective Bayesianism is recalled, based on the following three criteria: an epistemic criterion, a static coherence criterion and a dynamic coherence criterion. Then it is shown that (...) the adoption of this framework has two kinds of implications. The first one regards the methodology of the experimental study of probability judgment. The Bayesian framework creates pragmatic constraints on the methodology that are linked to the interpretation of, and the belief in, the information presented, or referred to, by an experimenter in order for it to be the basis of a probability judgment by individual participants. It is shown that these constraints have not been satisfied in the past, and the question of whether they can be satisfied in principle is raised and answered negatively. The second kind of implications consists of two limitations in the scope of the Bayesian model. They regard (1) the background of revision (the Bayesian model considers only revising situations but not updating situations), and (2) the notorious case of the null priors. In both cases Lewis’ rule is an appropriate alternative to Bayes’ rule, but its use faces the same operational difficulties. (shrink)

Information generally comes from less than fully reliable sources. Rationality, it seems, requires that one take source reliability into account when reasoning on the basis of such information. Recently, Bovens and Hartmann (2003) proposed an account of the conjunction fallacy based on this idea. They show that, when statements in conjunction fallacy scenarios are perceived as coming from such sources, probability theory prescribes that the “fallacy” be committed in certain situations. Here, the empirical validity of their model was assessed. The (...) model predicts that statements added to standard conjunction problems will change the incidence of the fallacy. It also predicts that statements from reliable sources should yield an increase in fallacy rates (relative to unreliable sources). Neither the former (Experiment 1) nor the latter prediction (Experiment 3) was confirmed, although Experiment 2 showed that people can derive source reliability estimates from the likelihood of statements in a manner consistent with the tested model. In line with the experimental results, model fits and sensitivity analyses also provided very little evidence in favor of the model. This suggests that Bovens and Hartmann’s present model fails to explain fully people’s judgements in standard conjunction fallacy tasks. (shrink)

My title is intended to recall Terence Fine's excellent survey, Theories of Probability [1973]. I shall consider some developments that have occurred in the intervening years, and try to place some of the theories he discussed in what is now a slightly longer perspective. Completeness is not something one can reasonably hope to achieve in a journal article, and any selection is bound to reflect a view of what is salient. In a subject as prone to dispute as this, there (...) will inevitably be many who will disagree with any author's views, and I take the opportunity to apologize in advance to all such people for what they will see as the narrowness and distortion of mine. (shrink)

A pure significance test would check the agreement of a statistical model with the observed data even when no alternative model was available. The paper proposes the use of a modified p -value to make such a test. The model will be rejected if something surprising is observed. It is shown that the relation between this measure of surprise and the surprise indices of Weaver and Good is similar to the relationship between a p -value, a corresponding odds-ratio, and a (...) logit or log-odds statistic. The s -value is always larger than the corresponding p -value, and is not uniformly distributed. Difficulties with the whole approach are discussed. (shrink)

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is (...) new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”. (shrink)

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference (...) class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains. (shrink)

Many philosophers deny that happiness can be equated with pleasurable experiences. Nozick introduced an experience machine thought experiment to support the idea that happiness requires pleasurable experiences that are “in contact with reality.” In this thought experiment, people can choose to plug into a machine that induces exclusively pleasurable experiences. We test Nozick’s hypothesis that people will reject this offer. We also contrast Nozick’s experience machine scenario with scenarios that are less artificial, and offer options which are less invasive or (...) disruptive than being connected to a machine, specifically scenarios in which people are offered an experience pill or a pill that improves overall functioning. (shrink)

The concepts of uncertainty in prediction and inference are introduced and illustrated using the diffraction of light as an example. The close relationship between the concepts of uncertainty in inference and resolving power is noted. A general quantitative measure of uncertainty in inference can be obtained by means of the so-called statistical distance between probability distributions. When applied to quantum mechanics, this distance leads to a measure of the distinguishability of quantum states, which essentially is the absolute value of the (...) matrix element between the states. The importance of this result to the quantum mechanical uncertainty principle is noted. The second part of the paper provides a derivation of the statistical distance on the basis of the so-called method of support. (shrink)